Revisiting Norm Estimation in Data Streams

We revisit the problem of (1±ε)-approximating the Lp norm, 0 ≤ p ≤ 2, of an n-dimensional vector updated in a stream of length m with positive and negative updates to its coordinates. We give several new upper and lower bounds, some of which are optimal. LOWER BOUNDS.We show that for the interesting range of parameters, Ω(ε log(nm)) bits of space are necessary for estimating Lp in one pass for any real constant p ≥ 0. If p is strictly positive, the lower bound improves to Ω(ε log(nmM)), where updates are in the set {−M, . . . ,M}. Our results imply space-optimality of the celebrated L2 estimation algorithm of [Alon, Matias, Szegedy JCSS 1999] and L1-difference algorithm of [Feigenbaum et al., SICOMP 2002], as well as a separation in the complexity of L0-estimation between the insertion-only and turnstile models since [Bar-Yossef et al., RANDOM 2002] give an Õ(ε + logn) space algorithm for the former. Our techniques also improve the best known lower bound for additive entropy approximation, and variants were used in [Clarkson, Woodruff, 2008] to obtain tight space bounds for streaming linear algebra problems. ALGORITHMS. We give the first one-pass streaming algorithm for estimating L0, the number of distinct elements in the turnstile model, in optimal space up to a constant factor. Our algorithm uses O(ε log(nmM)) space and has O(log(mM)) update time. For estimating Lp, 0 < p < 2, we improve the algorithm of [Indyk, J. ACM ’06] by reducing the space from O(ε log(nmM) log(n)) to O(ε log(nmM) log(n)/ log(1/ε)). In light of our lower bounds, this is optimal for any ε = n. We accomplish this by showing that a pseudorandom generator (PRG) construction of [Armoni, RANDOM 1998] can be improved by using a more space-efficient implementation of a recent extractor of [Guruswami, Umans, Vadhan, CCC 2007]. Specifically, the improved Armoni PRG stretches a seed ofO((S/(log(S)−log log(R)+O(1))) logR) bits to R bits fooling space-S algorithms for any R = 2, improving the O(S logR) seed length of the PRG of [Nisan, Combinatorica 1992], and improving several existing streaming algorithms. MIT Computer Science and Artificial Intelligence Laboratory. [email protected]. Supported by a National Defense Science and Engineering Graduate (NDSEG) Fellowship. Much of this work was done while the author was at the IBM Almaden Research Center. IBM Almaden Research Center, 650 Harry Road, San Jose, CA, USA. [email protected].